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American Mathematical Monthly - June/July 2005

Articles

Counting on Determinants
by Arthur T. Benjamin and Naiomi T. Cameron
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Many seemingly unrelated enumeration problems can be solved by determinants. These include the "determination" of nonintersecting paths in a network, the number of spanning trees of a graph, and the number of permutations with specified descent locations.

 

Continued Fractions of Tails of Hypergeometric Series
by Jonathan Michael Borwein, Kwok-Kwong Stephen Choi, and Wilfried Pigulla
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The tails of the Taylor series for many standard functions such as the arctangent and the logarithm can be expressed as continued fractions in a variety of ways. A surprising side effect is that some of these continued fractions provide a dramatic acceleration for the convergence of the underlying power series. In this article, we discuss three classes of continued fractions, namely, those of Euler, Gauss, and Perron, and study the convergence of the continued fractions of the hypergeometric series.

 

What is Just?
by Michel Balinski
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Equity in the sharing of things is an ever present concern, whether it regard taxes, pensions, inheritance, the allocation of costs and benefits among associates, apportioning seats among political parties or regions, or setting priorities among patients waiting for transplants of hearts, kidneys, or livers. Exactly how to share things found Aristotle and the Talmud expounding opposing views. A particularly enigmatic ruling of the Talmud, explained only some two millennia after the fact, shows the importance of a concept that is fundamental to all rules of fair division Â… and is but common sense. Known as "consistency" in the technical literature, a better word may be "coherence." It is the reason why what is just is so often confused with what is proportional.

 

Notes

A C Lagrange Inversion Theorem
by Nathaniel Grossman
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Mean Values and the Maximum Principle: A Proof in Search of More Theorems
by Mark A. Pinsky
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On the Perimeter of a Triangle in a Minkowski Plane
by H. Maehara and T. Zamfirescu
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A (K)-Ring Satisfying the Ascending Chain Condition On Principal Ideals That is not a Principal Ideal Ring
by Daniel Frohn
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Continuous Ranking of Zeros of Cylinder Functions
by Lee Lorch
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Evolution ofÂ…. The Isoperimetric Problem
By Viktor Blåsjö
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Problems and Solutions

Reviews

The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities.
By J. Michael Steele.
Reviewed by D. J. H. Garling
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Constantin Carathéodory: Mathematics and Politics in Turbulent Times.
By Maria Georgiadou.
Reviewed by Loukas Grafakos
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